Question 132787
Note: I'm <b>only</b> manipulating the left side. I'm showing the right just for comparison.





*[Tex \LARGE \frac{\cos \left(5x\right)\cos \left(3x\right)+\sin \left(5x\right)\sin \left(3x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Start with the given equation





*[Tex \LARGE \frac{\cos\left(5x-3x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Condense the numerator by using the identity {{{cos(x-y)=cos(x)cos(y)+sin(x)sin(y)}}}




*[Tex \LARGE \frac{\cos\left(2x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Combine like terms





*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Use the identity *[Tex \Large \cos\left(2x\right)=1-2\sin^2\left(x\right)] to replace the numerator




*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{\sin \left(2x+x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Expand 3x to 2x+x



*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{\sin\left(2x\right)\cos\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Expand {{{sin(2x+x)}}} through this identity *[Tex \Large \sin\left(x+y\right)=\sin\left(x\right)\cos\left(y\right)+\cos\left(x\right)\sin\left(y\right)]



*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\cos\left(x\right)\cos\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Replace {{{sin(2x)}}} with the identity *[Tex \Large \sin\left(2x\right)=2\sin\left(x\right)\cos\left(x\right)]



*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\cos^2\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Multiply





*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\left(1-\sin^2\left(x\right)\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Replace *[Tex \Large \cos^2\left(x\right)] with *[Tex \Large 1-\sin^2\left(x\right)]




*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-2\sin^3\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Distribute



*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-2\sin^3\left(x\right)+\left(1-2\sin^2\left(x\right)\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Replace *[Tex \Large \cos\left(2x\right)] with *[Tex \Large 1-2\sin^2\left(x\right)]




*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-2\sin^3\left(x\right)+\sin\left(x\right)-2\sin^3\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}] Distribute




*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-4\sin^3\left(x\right)}=\frac{\csc\left(x\right)}{2}]  Combine like terms





*[Tex \LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\left(1-2\sin^2\left(x\right)\right)}=\frac{\csc\left(x\right)}{2}] Factor out the GCF *[Tex \Large 2\sin\left(x\right)]




*[Tex \LARGE \frac{1}{2\sin\left(x\right)\left(1\right)}=\frac{\csc\left(x\right)}{2}] Cancel like terms







*[Tex \LARGE \frac{1}{2\sin\left(x\right)}=\frac{\csc\left(x\right)}{2}] Simplify 




*[Tex \LARGE \frac{\csc\left(x\right)}{2}=\frac{\csc\left(x\right)}{2}] replace {{{1/sin(x)}}} with {{{csc(x)}}}